didericis
d87b94d3b3
Adds Definition (flank face) + 3 lemmas + a partial theorem proving
Conjecture (Deciding face) -- hence Conjecture 5.1 -- for the case
where at least one neighbour of v in the parent triangulation G has
degree ≤ 6:
- Lemma (Flank-length formula): |F_{i, i+1}^♭| = n_i - 1.
- Lemma (Flank covering, n_i = 5): boundary of F_{i, i+1}^♭ is in
V(K_b) ∪ V(K_c). Proof: the single intermediate P is adjacent to
both A_i and A_{i+1}; A_{i+1} ∈ V(K_b) ∩ V(K_c) via spike, so
A_{i+1}'s c_0-edge (in K_b) or c_1-edge (in K_c) lands on P.
- Lemma (Flank covering, n_i = 6): two intermediates P_1, P_2; P_2
handled as the n_i = 5 case; P_1 covered by case analysis on
φ(A_i P_1) ∈ {c, c_1}: in Case (a) K_b walks A_i → P_1 directly;
in Case (b) propagation from P_2 via P_2's c-edge to P_1 (forced
by properness at P_1 ruling out φ(P_1 P_2) = c_1).
- Theorem (Partial proof of Conjecture (Deciding face)): combining
the length formula and the covering lemmas, F_{i, i+1}^♭ is a
deciding face whenever n_i ∈ {5, 6}, since its length is then 4
or 5 (≢ 0 mod 3).
The remaining structural case is n_i ≥ 7 for all i (= all five
neighbours of v in G have degree ≥ 7); for these the merged-side
face F^♭_{i+3, i+4} of length n_{i+3} - 2 often plays the deciding
role empirically, but a uniform structural argument is left open.
Paper grows from 18 to 20 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
math-research
Personal mathematics research repository by Eric Bauerfeld. Papers are written in AMS-LaTeX using the amsart document class.
Papers
kempe_style_search_for_smaller_contradiction
Humans Suffice: A Novel Proof of the Four Color Theorem
An in-progress proof of the Four Color Theorem via a minimal counterexample argument. The paper builds on Kempe's 1879 strategy — establishing valid cases for vertices of degree ≤ 4, then extending the argument to the degree-5 case using properties of non-adjacent degree-5 vertices, merged subgraphs, and locked colorings.
plane_depth_labelling
Plane Depth Labelling
Early-stage paper. Title and author information set; content in progress.
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